Yes, there is still a legitimate way to ask Apple to remove Activation Lock from your gifted M1 MacBook Air, even if the previous owner has forgotten her Apple Account email address and password.
However, removal is not guaranteed, because Apple must be satisfied that the device is eligible and that the ownership documentation is acceptable.
The first thing I would recommend is helping your friend recover her Apple Account. She can visit Apple's account recovery website using any web browser. She should try any email addresses or phone numbers she may have used when setting up the Mac. Checking old email accounts for messages from Apple, purchase receipts, or account notifications may also help her identify the correct email address. If she remembers the email address but has forgotten the password, she can follow Apple's password-reset process. Apple's guide to recovering an Apple Account provides further instructions.
If account recovery is unsuccessful, your next step should be to submit an official Activation Lock support request through Apple's Activation Lock support portal. Apple has a process for reviewing requests to remove Activation Lock when the requester has appropriate proof of purchase. You can also consult Apple's official guides on removing Activation Lock and Activation Lock for Mac.
Your friend should gather as much documentation as possible before submitting the request. The strongest evidence is generally the original purchase receipt or invoice identifying the Mac, ideally including its serial number. If she purchased it secondhand, she should provide any legitimate purchase documentation she still has. Since she gifted the Mac to you, a signed statement confirming that she owned the computer and gave it to you may also help explain the circumstances. However, a gift statement alone may not satisfy Apple's proof-of-purchase requirements, and Apple will decide whether the available evidence is sufficient. The Mac's serial number may be found on the device itself, its original packaging, or the purchase documentation.
You can also contact Apple Support or visit an Apple Store to explain the situation and ask what documentation would be acceptable. An in-person visit does not guarantee that Activation Lock will be removed, but Apple can clarify the available options for your particular case.
Unfortunately, wiping the SSD, reinstalling macOS, or restoring the Mac using DFU mode will not remove Activation Lock. The lock is associated with Apple's activation servers, so restoring the computer does not clear its association with the previous owner's account. I would also avoid third-party unlocking services that promise a guaranteed permanent solution, as these may be scams or offer unreliable bypasses.
If Apple cannot verify the ownership and your friend cannot recover her account, the Mac may remain Activation Locked. In that situation, there is no guaranteed legitimate workaround, and replacing the logic board would generally be an expensive and potentially uneconomical option.
My recommendation is to try account recovery first, then submit an official Activation Lock request with any purchase documentation your friend can provide.
You are not necessarily stuck with an unusable Mac, but the outcome depends largely on whether Apple can verify the device's ownership.
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